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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

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[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
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April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
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Fixed angle
iv999xyz   0
4 minutes ago
Let ABC be an acute triangle with circumcircle w and D is fixed point on BC. E is randomly selected on BC and C, D and E are in this order. AE intersects w again at F and circumcircle around DEF intersects w again at M. Prove that ∠ABM is fixed.
0 replies
iv999xyz
4 minutes ago
0 replies
J
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6 tangents to 1 circle
moony_   1
N 17 minutes ago by soryn
Source: own
Let $P$ be a point inside the triangle $ABC$. $AP$ intersects $BC$ at $A_0$. Points $B_0$ and $C_0$ are defined similarly. Line $B_0C_0$ intersects $(ABC)$ at points $A_1$, $A_2$. The tangents at these points to $(ABC)$ intersect BC at points $A_3$, $A_4$. Points $B_3$, $B_4$, $C_3$, $C_4$ are defined similarly. Prove that points $A_3$, $A_4$, $B_3$, $B_4$, $C_3$, $C_4$ lie on one conic
1 reply
moony_
Yesterday at 9:30 AM
soryn
17 minutes ago
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2024 numbers in a circle
PEKKA   30
N 18 minutes ago by NicoN9
Source: Canada MO 2024/2
Jane writes down $2024$ natural numbers around the perimeter of a circle. She wants the $2024$ products of adjacent pairs of numbers to be exactly the set $\{ 1!, 2!, \ldots, 2024! \}.$ Can she accomplish this?
30 replies
PEKKA
Mar 8, 2024
NicoN9
18 minutes ago
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Inspired by ZDSX 2025 Q845
sqing   5
N 43 minutes ago by sqing
Source: Own
Let $ a,b,c>0 $ and $ a^2+b^2+c^2 +ab+bc+ca=6 $ . Prove that$$ \frac{1}{2a+bc }+ \frac{1}{2b+ca }+ \frac{1}{2c+ab }\geq 1$$
5 replies
sqing
Yesterday at 1:41 PM
sqing
43 minutes ago
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No more topics!
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J
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pure geometry(prove)
Pirkuliyev Rovsen   1
N Aug 30, 2014 by sunken rock
Let $ABCD$ be a quadrilateral inscribed in a circle of diameter $BD$ and $M$ be an arbitrary point on the shorter arc $AD$.Let $MN$, $MK$, $MP$, $MT$ be perpendiculars from $M$ onto lines $AB$, $BC$, $CD$, $AD$ respectively.Prove that $S_{MNP}=S_{MKT}$.
1 reply
Pirkuliyev Rovsen
Aug 30, 2014
sunken rock
Aug 30, 2014
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pure geometry(prove)
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Pirkuliyev Rovsen
5047 posts
Pirkuliyev Rovsen
#1 h  Aug 30, 2014, 5:30 AM • 2 Y
Y by Adventure10, Mango247
Let $ABCD$ be a quadrilateral inscribed in a circle of diameter $BD$ and $M$ be an arbitrary point on the shorter arc $AD$.Let $MN$, $MK$, $MP$, $MT$ be perpendiculars from $M$ onto lines $AB$, $BC$, $CD$, $AD$ respectively.Prove that $S_{MNP}=S_{MKT}$.
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sunken rock
4381 posts
sunken rock
#2 h  Aug 30, 2014, 6:08 AM • 3 Y
Y by Pirkuliyev Rovsen, Adventure10, Mango247
$\angle KMT+\angle NMP=180^\circ\ (\ 1\ )$. $MPDP, MNBK, ABDM$ being cyclic give $\triangle MTP\sim\triangle MNK\implies\frac{MT}{MP}=\frac{MN}{MK}\ (\ 2\ )$. From $(1)$ and $(2)$ we are done.

Best regards,
sunken rock
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